The Topologically Transitivity Of Some Hyperspace Dynamical Systems

The paper consists of three chapter .In the first chapter , we introduce the basic concept of dynamical system and chaos ,and the result until now on transitivity of linear chaos and hypercyclic operators.In the second chapter, we study separability of some hyperspace of Banach space. And we get that if the Banach space is separable, then its hyperspace W_k ( X ) and hyperspace W_kc( X ) are also separable. Using James theorem, we prove that the natural extension of the bounded linear operator on Banach space is continuous on hyperspace such as W_k ( X )and W_kc( X ).We investigate the relationship between the chaoticity of some hyperspace set-valued discrete dynamical systems ( W_k ( X ), f ) ,(K_k( X ), f ), ( W_kc( X ), f ) and the chaoticity of linear dynamical system ( X ,f )on the Banach space X . At last we get if f :X→Xis bounded linear operator on the separable Banach space X , then the f satisfies the Hypercyclicity Criterion is equivalent with that the set-valued dynamical systems ( W_k ( X ), f ), ( K_k( X ), f ), ( W_kc( X ), f ) is transitiveAt the last chapter, we make a research on the relationship between the chaoticity of compact metric space and the chaoticity of Vietories topology, Vietories Upper topology and Wijsman topology on its hyperspace K ( X ).